Floating Point Numbers

Abstract

Numbers from $\mathbb {R}$ cannot be handled directly by computers, as they can have infinitely many digits after the decimal point. We have to truncate their representation. Storage schemes with a finite number of digits before and after the decimal point struggle: There are always regimes of values where they work prefectly fine and regimes where they cannot represent values accurately. We therefore introduce the concept of floating point numbers and discuss the IEEE standard. Its concept of normalisation allows us to establish the notion of machine precision, as we realise that the truncation into a finite number of digits equals rounding and introduces a bounded relative error. Since we work with approximations to real numbers, we have to be careful when we compare floating point numbers in our codes.